Question

xcube + 8ycube - 64zcube +24xyz
Hint: use property ancube +b cube +c cube = 3abc

Question is of class 9

Answer 1

Answer:

Step-by-step explanation:

aking RHS of the identity:  

(a + b + c)(a2 + b2 + c2 - ab - bc - ca )  

Multiply each term of first polynomial with every term of second polynomial, as shown below:  

= a(a2 + b2 + c2 - ab - bc - ca ) + b(a2 + b2 + c2 - ab - bc - ca ) + c(a2 + b2 + c2 - ab - bc - ca )  

= { (a X a2) + (a X b2) + (a X c2) - (a X ab) - (a X bc) - (a X ca) } + {(b X a2) + (b X b2) + (b X c2) - (b X ab) - (b X bc) - (b X ca)} + {(c X a2) + (c X b2) + (c X c2) - (c X ab) - (c X bc) - (c X ca)}  

Solve multiplication in curly braces and we get:  

= a3 + ab2 + ac2 - a2b - abc - a2c + a2b + b3 + bc2 - ab2 - b2c - abc + a2c + b2c + c3 - abc - bc2 - ac2  

Rearrange the terms and we get:  

= a3 + b3 + c3 + a2b - a2b + ac2- ac2 + ab2 - ab2 + bc2 - bc2 + a2c - a2c + b2c - b2c - abc - abc - abc  

Above highlighted like terms will be subtracted and we get:  

= a3 + b3 + c3 - abc - abc - abc  

Join like terms i.e (-abc) and we get:  

= a3 + b3 + c3 - 3abc  

Hence, in this way we obtain the identity i.e. a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)  

Let's try the following examples of this identity  

Example: Solve 8a3 + 27b3 + 125c3 - 30abc  

Solution: This proceeds as:  

Given polynomial (8a3 + 27b3 + 125c3 - 30abc) can be written as:  

(2a)3 + (3b)3 + (5c)3 - (2a)(3b)(5c)  

And this represents identity:  

a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)  

Where a = 2a, b = 3b and c = 5c  

Now apply values of a, b and c on the L.H.S of identity i.e. a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca) and we get:  

= (2a + 3b + 5c) { (2a)2 +(3b)2 + (5c)2 - (2a)(3b) - (3b)(5c) - (5c)(2a) }  

Expand the exponential forms and we get:  

= (2a + 3b + 5c) {4a2 +9b2 + 25c2 - (2a)(3b) - (3b)(5c) - (5c)(2a) }  

Solve brackets and we get:  

= (2a + 3b + 5c) {4a2 +9b2 + 25c2 - 6ab - 15bc - 10ca}  

Hence, 8a3 + 27b3 + 125c3 - 30abc = (2a + 3b + 5c) (4a2 + 9b2 + 25c2 - 6ab - 15bc - 10ca)  

Answer 2

hey friend your answer↓

(a+b) ^3=a^3+3a^2b+3ab^2+b^3